\newproblem{lay:3_3_26}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 3.3.26}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $T:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be a linear transformation, and let $\mathbf{p}$ be a vector and $S$ a set in $\mathbb{R}^n$. Show that the image of
	$\mathbf{p}+S$ under $T$ is the translated set $T(\mathbf{p})+T(S)$ in $\mathbb{R}^m$.
}{
   % Solution
	Any vector of the set $\mathbf{p}+S$ is of the form
	\begin{center}
		$\mathbf{x}=\mathbf{p}+\mathbf{s}$
	\end{center}
	where $\mathbf{s}\in S$. If we apply $T$ to $\mathbf{x}$ and exploiting the fact that $T$ is a linear transformation, we get
	\begin{center}
		$T(\mathbf{x})=T(\mathbf{p}+\mathbf{s})=T(\mathbf{p})+T(\mathbf{s})$
	\end{center}
	The set of all vectors of the form $T(\mathbf{s})$ is actually $T(S)$, so we have that, as stated by the problem,
	\begin{center}
		$T(\mathbf{x})\in T(\mathbf{p})+T(S)$
	\end{center}
}
\useproblem{lay:3_3_26}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
